Rhombitetrahexagonal tiling
Rhombitetrahexagonal tiling | |
---|---|
Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | 4.4.6.4 |
Schläfli symbol | rr{6,4} or |
Wythoff symbol | 4 | 6 2 |
Coxeter diagram | |
Symmetry group | [6,4], (*642) |
Dual | Deltoidal tetrahexagonal tiling |
Properties | Vertex-transitive |
In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.
Constructions
There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1+,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).
Name | Rhombitetrahexagonal tiling | |
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Image | ||
Symmetry | [6,4] (*642) | [6,1+,4] = [∞,3,∞] (*3222) = |
Schläfli symbol | rr{6,4} | t0,1,2,3{∞,3,∞} |
Coxeter diagram | = |
There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with [6,4+] (4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with [6+,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6+,4+] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.
Lower symmetry constructions | |||||||||||
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[6,4], (*632) | [6,4+], (4*3) | ||||||||||
[6+,4], (6*2) | [6+,4+], (32×) |
This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of .
Symmetry
The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1+,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.
Related polyhedra and tiling
*n42 symmetry mutation of expanded tilings: n.4.4.4
| |||||||||||
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Symmetry [n,4], (*n42) | Spherical | Euclidean | Compact hyperbolic | Paracomp. | |||||||
*342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4] | *∞42 [∞,4] | |||||
Expanded figures | |||||||||||
Config. | 3.4.4.4 | 4.4.4.4 | 5.4.4.4 | 6.4.4.4 | 7.4.4.4 | 8.4.4.4 | ∞.4.4.4 | ||||
Rhombic figures config. | V3.4.4.4 | V4.4.4.4 | V5.4.4.4 | V6.4.4.4 | V7.4.4.4 | V8.4.4.4 | V∞.4.4.4 |
Uniform tetrahexagonal tilings
| |||||||||||
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Symmetry: [6,4], (*642) (with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (*3232) index 4 subsymmetry) | |||||||||||
= = = | = | = = = | = | = = = | = | ||||||
{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
Uniform duals | |||||||||||
V64 | V4.12.12 | V(4.6)2 | V6.8.8 | V46 | V4.4.4.6 | V4.8.12 | |||||
Alternations | |||||||||||
[1+,6,4] (*443) | [6+,4] (6*2) | [6,1+,4] (*3222) | [6,4+] (4*3) | [6,4,1+] (*662) | [(6,4,2+)] (2*32) | [6,4]+ (642) | |||||
= | = | = | = | = | = | ||||||
h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |
Uniform tilings in symmetry *3222
| ||||
---|---|---|---|---|
64 | 6.6.4.4 | (3.4.4)2 | 4.3.4.3.3.3 | |
6.6.4.4 | 6.4.4.4 | 3.4.4.4.4 | ||
(3.4.4)2 | 3.4.4.4.4 | 46 |
See also
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch